Algebraic properties of cryptosystem PGM

In the late 1970s Magliveras invented a private-key cryptographic system calledPermutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, calledlogarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space ℤ_|G| for a given finite permutation groupG. In this paper we prove several algebraic properties of PGM. We show that the set of PGM transformations ℐ _G is not closed under functional composition and hence not a group. This set is 2-transitive on ℤ_|G| if the underlying groupG is not hamiltonian and not abelian. Moreover, if the order ofG is not a power of 2, then the set of transformations contains an odd permutation. An important consequence of these results is that the group generated by the set of transformations is nearly always the symmetric group ℒ_|G|. Thus, allowing multiple encryption, any permutation of the message space is attainable. This property is one of the strongest security conditions that can be offered by a private-key encryption system. S. S. Magliveras was supported in part by NSF/NSA Grant Number MDA904-82-H0001, by U.S. West Communications, and by the Center for Communication and Information Science of the University of Nebraska.     

Rn 上的线性局部保控映射

设φ：Rn →Rn 是一个线性映射,对任意x∈Rn ,都存在与x有关的线性保控映射ψx ,使得φ（x）＝ψx（x）,则称φ是欧氏空间上的一个线性局部保控映射。线性保控映射是线性局部保控映射,其逆命题不成立。文中定义了两个与φ相关的Rn的子集Iφ和IIφ,如果任意Rn上的点都属于Iφ（或IIφ）,那么这个线性局部保控映射φ是线性保控映射。     

REMARKS ON JOHN DISKS

Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.     

Continuity versus Discretization

The threefold interest in architecture, biology and mathematics motivated us to examine and justify new architectural forms. We discuss some notions of rhythm: Euclidean, morphogenetic and morphologic. Contemporary relationships between structure and form are based on the generation of shape by technological processes, thus the resulting objects are restricted to their material expression. Here a phenomenological organisation of form and its continuity with the landscape arise out of the mathematical and architectural creativity. The use of the computer is applied from outside to inside instead from inside to outside; this means that we are dealing with the organisational processes via continuous methods instead of evolutionary processes given by computer simulations, known as genetic algorithms, where the resulting configurations are reduced to a matter of routine. The role of design as an aesthetic component innovates the theoretical framework of structural engineering to establish the architectural environments.     

广义混合均衡与不动点问题公共解的迭代算法

在Hilbert空间中,引入一种新的关于广义混合均衡问题的迭代方法,通过该迭代方法求得广义混合均衡问题的解集与非扩张映像的不动点集的公共解.并且在一定条件下,得到强收敛定理.所得结果改进和推广了这一领域内某些最新结果.     

Nonlinear Inverse Boundary-Value Problems of Conformal Mapping with a Controlling Potential

Some characteristic cases of formation of an ideal flow are described depending on preset values of controlling potential. Based on the modified G. Polozhii's method of summary mappings, the characteristic function of the flow is obtained. Examples of construction of a dynamic grid, field of velocities, and formation of different flows are given. Types of key problems of finding the values of controlling potential, which provide optimization of definite functionals are defined. An example of solution to one of them is presented.     

新疆西准噶尔哈图岩体地球化学特征及其地质意义

哈图岩体侵位于西准噶尔南部早—中泥盆世海相火山-沉积地层中,主要岩性为二长花岗岩、正长花岗岩及少量碱长花岗岩,是研究西准噶尔晚石炭世构造环境的理想对象。哈图岩体花岗岩中SiO₂含量(质量分数,下同)为72.06%～76.53%,全碱(Na₂O+K₂O)为7.80%～8.57%,CaO为0.20%～1.58%,MgO为0.07%～0.42%,明显具有高Si、富碱、贫Ca、低Mg的特征; 这些花岗岩的Al₂O₃含量为12.55%～14.06%,A/CNK值为1.06～1.15,A/NK值为1.15～1.36,显示其为弱过铝质—强过铝质花岗岩。此外,哈图岩体具有较低的稀土元素总含量((88.14～139.53)×10^-6)和高的轻、重稀土元素比值((La/Yb)_N值为3.04～6.50),在球粒陨石标准化稀土元素配分模式中显示为典型的右倾“V”字形且具有强烈的负Eu异常,表明源区可能有斜长石残留或出现斜长石的分离结晶作用; 在原始地幔标准化微量元素蛛网图中,这些花岗岩明显富集Rb、Th、U、K、Pb等大离子亲石元素,相对富集Zr、Hf,强烈亏损Sr、Ti、P,相对亏损Ba、Nb。以上特征及主量、微量元素特征和图解表明,哈图岩体为典型的弱过铝质—强过铝质A₂型花岗岩,其形成于造山后伸展阶段,是由洋壳和岛弧组成的新生地壳受上涌的幔源物质加热,在浅部低压条件下发生部分熔融,再经分离结晶作用的产物。     

Common fixed points of a countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces

In this paper, we introduce a one-step iterative scheme for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mappings in a real uniformly convex Banach space. We establish weak and strong convergence theorems of the proposed iterative scheme under some control conditions.     

Function evaluation with feedforward neural networks

Much effort has previously been spent in investigating the decision making/object identification capabilities of feedforward neural networks. In the present work we examine the less frequently investigated abilities of such networks to implement computationally useful operations in arithmetic and function evaluation. The approach taken is to employ standard training methods, such as backpropagation, to teach simple three-level networks to perform selected operations ranging from one-to-one mappings to many-to-many mappings. Examples considered cover a wide range, such as performing reciprocal arithmetic on real valued inputs, implementing particle identifier functions for identification of nuclear isotopes in scattering experiments, and locating the coordinates of a charged particle moving on a surface. All mappings are required to interpolate and extrapolate from a small sample of taught exemplars to the general continuous domain of possible inputs. A unifying principle is proposed that looks upon all such function constructions as expansions in terms of basis functions, each of which is associated with a hidden node and is parameterized by such techniques as gradient descent methods.